On the Connectivity of Fiber Graphs

We consider the connectivity of fiber graphs with respect to Gr\"obner basis and Graver basis moves. First, we present a sequence of fiber graphs using moves from a Gr\"obner basis and prove that their edge-connectivity is lowest possible and can have an arbitrarily large distance from the minimal degree. We then show that graph-theoretic properties of fiber graphs do not depend on the size of the right-hand side. This provides a counterexample to a conjecture of Engstr\"om on the node-connectivity of fiber graphs. Our main result shows that the edge-connectivity in all fiber graphs of this counterexample is best possible if we use moves from Graver basis instead.